I'm learning Castelnuovo-Mumford regularity of associated graded ring. There is a lemma:
It's from "Castelnuovo-Mumford regularity postulation number and relation types" by Markus Brodmann and Cao Huy Linh.
Now, I want to extend it to modules.
Lemma 3.3. Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension one. $M$ is an $A$-module with $\operatorname{dim} M=1$, $I=(x)$ is a submodule of $M$. Then
$$\operatorname{reg}(G_I(M))=p(G_I(M))+\operatorname{depth}(A).$$
In proof, I also replace from above proof: $\operatorname{depth}(A)$ to $\operatorname{depth}(M)$, $\mathfrak{q}$ to $I$, $L:=H^0_{\mathfrak{m}}(M)$,...
So, please show me wrong point in new proof. Please do it with more detail. Thank so much!